How do you simplify #sqrt205#?

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Answer 1 · 2018-05-19T19:27:36

#sqrt(205)# is already in simplest form.

Note that:

#205 = 5 * 41#

has no square factors larger than #1#. As a result, its square root cannot be simplified.

Using the algorithm described in https://socratic.org/s/aR4mTUXq we can find that the continued fraction expansion for #sqrt(205)# is:

#sqrt(205) = [14;bar(3,6,1,4,1,6,3,28)]#

#color(white)(sqrt(205)) = 14+1/(3+1/(6+1/(1+1/(4+1/(1+1/(6+1/(3+1/(28+...))))))))#

From this we can find the smallest solution of Pell's equation for #205#:

#39689^2 = 205 * 2772^2 + 1#

and related efficient rational approximation:

#sqrt(205) ~~ 39689/2772 ~~ 14.31782107#